SOLUTION: Cynthia Besch wants to buy a rug for a room that is 23 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 408 square feet of

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: Cynthia Besch wants to buy a rug for a room that is 23 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 408 square feet of      Log On

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Question 1113179: Cynthia Besch wants to buy a rug for a room that is 23 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 408
square feet of carpeting. What dimensions should the rug​ have?
Found 2 solutions by josmiceli, Theo:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
The dimensions and area of the rug are:
+%28+23+-+2x+%29%2A%28+30+-+2x+%29+=+408+
+690+-+60x+-+46x+%2B+4x%5E2+=+408+
+4x%5E2+-+106x+%2B+282+=+0+
+2x%5E2+-+53x+%2B+141+=+0+
Use quadratic formula
+x+=+%28+-b+%2B_sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282a%29+
+a+=+2+
+b+=+-53+
+c+=+141+
+x+=+%28+-%28-53%29+%2B-sqrt%28+%28-53%29%5E2+-+4%2A2%2A141+%29%29%2F4+
+x+=+%28+53+%2B-sqrt%28+2809+-+1128+%29%29+%2F+4+
+x+=+%28+53+%2B+sqrt%28+1681+%29%29+%2F+4%0D%0A%7B%7B%7B+x+=+%28+53+%2B+41+%29%2F4+
+x+=+94%2F4+
+x+=+23.5+ ( too large for the border )
Use the neg square root
+x+=+%28+53+-+41+%29%2F4+
+x+=+12%2F4+
+x+=+3+
and
+23+-+2x+=+23+-+6+
+23+-+2x+=+17+
and
+30+-+2x+=+30+-+6+
+30+-+2x+=+24+
------------------------------
The rug is 17 x 24
------------------------------
check:
+17%2A24+=+408+
OK

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
let the distance between the rug and the walls be x on all sides.

the area of the room is 30 * 23 = 690

the area of the rug is (30 - 2x) * (23 - 2x) = 408

perform the multiplication to get:

690 - 60x - 46x + 4x^2 = 408

combine like terms to get:

690 - 106x + 4x^2 = 408

subtract 408 from both sides of the equation to get:

690 - 106x + 4x^2 - 408 = 0

combine like terms to get:

282 - 106x + 4x^2 = 0

reorder the terms in descending order of degree to get:

4x^2 - 106x + 282 = 0

factor this quadratic equation to get:

x = 23.5 or x = 3

x can't be 23.5 because then the area of the rug would not be equal to 408 because:

(30 - 2 * 23.5) * (23 - 2 * 23.5) equals a negative.

x has to be 3 or none of the possible solutions are good.

when x = 3, you get (30 - 6) * (23 - 6) = 408.

this results in 24 * 17 = 408 which becomes 408 = 408 which is true.

your solution is that x = 3 and the dimensions of the rug are 24 by 17.

this rug fits into a room that is 30 by 23.

there will be a border around the rug that is 3 feet in all directions.

the dimensions of the room are 30 by 23

the dimensions of the rug are 24 by 17.

the difference between the length of the room and the length of the rug is 30 - 24 = 6.

the difference between the width of the room and the width of the rug is 23 17 = 6.

divide this difference by 2 and you get 3 foot border between the rug and the wall all around.

here's my picture.

$$$